ON THE COMPLEXITY OF THE SET OF
UNCONDITIONAL CONVEX BODIES
MARK RUDELSON
Abstract. We show that for any 1 ≤ t ≤ c̃n1/2 log−5/2 n, the set
of unconditional convex bodies in Rn contains a t-separated subset
of cardinality at least
c
n
.
exp exp 2
t log4 (1 + t)
This implies the existence of an unconditional convex body in Rn
which cannot be approximated within the distance d by a projection of a polytope with N faces unless N ≥ exp(c(d)n).
We also show that for t ≥ 2, the cardinality of a t-separated set
of completely symmetric bodies in Rn does not exceed
log2 n
exp exp C
.
log t
1. introduction
In [1] Barvinok and Veomett posed a question whether any n-dimensional convex symmetric body can be approximated by a projection of
a section of a simplex whose dimension is subexponential in n. The
importance of this question stems from the fact that the convex bodies
generated this way allow an efficient construction of the membership
oracle. The question of Barvinok and Veomett has been answered
in [4], where it was shown that for all 1 ≤ n ≤ N , there exists
an n-dimensional symmetric convex body B such that for every ndimensional convex body K obtained as a projection of a section of an
N -dimensional simplex one has
s
n
d(B, K) ≥ c
,
2N ln(2N )
ln
n
where d(·, ·) denotes the Banach-Mazur distance and c is an absolute
positive constant. Moreover, this result is sharp up to a logarithmic
factor.
Department of Mathematics, University of Michigan. Partially supported by
NSF grants DMS-01161372, DMS-1464514, and USAF Grant FA9550-14-1-0009.
1
2
MARK RUDELSON
One of the main steps in the proof of this result was an estimate of
the complexity of the set of all convex symmetric bodies in Rn , i. e., the
Minkowski or Banach–Mazur compactum. The complexity is measured
in terms of the maximal size of a t-separated set with respect to the
Banach–Mazur distance
d(K, D) = inf{λ ≥ 1 | D ⊂ T K ⊂ λD},
where the infimum is taken over all linear operators T : Rn → Rn .
A set A in a metric space (X, d) is called t-separated if the distance
between any two distinct points of A is at least t. It follows from [4]
that for any 1 ≤ t ≤ cn, the set of all n-dimensional convex bodies
contains a t-separated subset of cardinality at least
(1.1)
exp(exp(cn/t)).
More precisely, Theorem 2.3 [4] asserts that for any 2n ≤ M ≤ en ,
there exists a probability measure P M on the set of convex symmetric
polytopes such that
cn
0
00
0
00
(1.2) P M ⊗ P M
(K , K ) | d(K , K ) ≤
≤ 2e−nM .
ln(M/n)
This probabilistic estimate together with the union bound implies the
required lower bound on the maximal size of a t-net.
Note that for t = O(1), the estimate above shows that the complexity of the Minkowski compactum is doubly exponential in terms of the
dimension. This fact has been independently established by Pisier [6],
who asked whether a similar statement holds for the set of all unconditional convex bodies and for the set of all completely symmetric bodies.
We show below that the answer to the first question is affirmative, and
to the second one negative.
Consider unconditional convex bodies first. A convex symmetric
body K ⊂ Rn is called unconditional if it symmetric with respect to
all coordinate hyperplanes. This property can be conveniently reformulated in terms of the norm generated by K. For x ∈ Rn , set
kxkK = min{a ≥ 0 | x ∈ aK}.
P
The body K is unconditional if for any x = nj=1 xj ej , and for any
J ⊂ [n],
X
X
,
xj ej ≤ x
e
j
j
j∈J
j∈[n]
K
K
i.e., whenever all coordinate projections are contractions.
COMPLEXITY OF THE SET OF UNCONDITIONAL CONVEX BODIES
3
Our main result shows that the complexity of the set Knunc of unconditional convex bodies at the scale t is doubly exponential as long as
t = O(1). More precisely, we prove the following theorem.
Theorem 1.1. Let 1 ≤ t ≤ c̃n1/2 log−5/2 n. The set of n-dimensional
unconditional convex bodies contains a t-separated set of cardinality at
least
c
exp exp 2 4
n
.
t log (1 + t)
Here, c̃ and c are positive absolute constants.
Note that unlike the estimate (1.1), which is valid for 1 ≤ t ≤ cn,
the estimate above holds only in the range 1 ≤ t ≤ c̃n1/2 log−5/2 n. By
a theorem of Lindenstrauss and Szankowski [3], the maximal Banach–
Mazur distance between two n-dimensional unconditional bodies does
not exceed Cn1−ε0 for some ε0 ≥ 1/3. This means that a non-trivial
estimate of the cardinality of a t-separated set in Knunc is impossible
whenever t > n1−ε0 .
Following the derivation of Theorem 1.1 [1], one can show that Theorem 1.1 implies a result on the hardness of approximation of an unconditional convex body by a projection of a section of a simplex refining
the solution of the problem posed by Barvinok and Veomett.
Corollary 1.2. Let n ≤ N . There exists an n-dimensional unconditional convex body B, such that for every n-dimensional convex body K
obtained as a projection of a section of an N -dimensional simplex one
has
1/4
n
n
−1
d(B, K) ≥ c
· log
,
log N
log N
where c is an absolute positive constant.
In particular, Corollary 1.2 means that to be able to approximate
all unconditional convex bodies in Rn by projections of sections of
an N -dimensional simplex within the distance O(1), one has to take
N ≥ exp(cn).
Consider now the set of completely symmetric bodies. We will call an
n-dimensional convex body completely symmetric if it is unconditional
and invariant under all permutations of the coordinates. This term
is not commonly used. In the language of normed spaces, completely
symmetric convex bodies correspond to the spaces with 1-symmetric
basis. However, since the term “symmetric convex bodies” has a different meaning, we will use “completely symmetric” for this class of
bodies.
4
MARK RUDELSON
The set of completely symmetric convex bodies is much smaller than
the set of all unconditional ones. This manifests quantitatively in the
fact that the cardinality of a t-separated set of completely symmetric
bodies is significantly lower. Namely, we prove the following proposition in Section 5.
Proposition 1.3. Let t ≥ 2. The cardinality of any t-separated set in
Kcs does not exceed
log2 n
exp exp C
.
log t
This proposition means, in particular, that the complexity of the set
of completely symmetric convex bodies is not doubly exponential in
the dimension, which answers the second question of Pisier.
Acknowledgement. The author is grateful to Olivier Guédon for several suggestions which allowed to clarify the presentation.
2. Notation and an outline of the construction
Let us list some basic notation used in the proofs below. By P and
E we denote the probability and the expectation. If N is a natural
number, then [N ] denotes the set of all integers from 1 to N .
Let 1 ≤ p ≤ ∞. By kxkp we denote the standard `p -norm of a vector
x = (x1 , . . . , xn ) ∈ Rn :
!1/p
n
X
kxkp =
|xj |p
,
j=1
and Bpn denotes the unit ball of `np . If A : Rn → Rm is a linear operator,
and K1 , K2 are convex symmetric bodies, then kA : K1 → K2 k stands
for the operator norm of A considered as an operator between normed
spaces with unit balls K1 and K2 :
kA : K1 → K2 k = max kAxkK2 .
x∈K1
The norm kA : B2n → B2m k is denoted simply by kAk. The Hilbert–
Schmidt norm of A is
!1/2
m X
n
X
kAkHS =
|aij |2
.
i=1 j=1
COMPLEXITY OF THE SET OF UNCONDITIONAL CONVEX BODIES
5
For K1 , . . . , KL ⊂ Rn , we denote by abs.conv(K1 , . . . , KL ) their absolute convex hull
L
L
X
X
abs.conv(K1 , . . . , KL ) = {
λ l xl |
|λl | ≤ 1, xl ∈ Kl for l ∈ [L]}.
l=1
l=1
Also, we define the unconditional convex hull of the points x1 , . . . , xL ∈
Rn with coordinates xj = (xj1 , . . . , xjn ) by
unc.conv(x1 , . . . , xL ) = conv (ε11 x11 , . . . , ε1n x1n ), . . . , (εL1 xL1 , . . . , εLn xLn ) ,
where the convex hull is taken over all choices of εli ∈ {−1, 1}. Obviously, unc.conv(x1 , . . . , xL ) is the smallest unconditional convex set
containing x1 , . . . , xL .
Finally, C, c, c0 etc. denote absolute constants whose value may
change from line to line.
The random convex bodies K 0 , K 00 appearing in (1.2) are generalized
Gluskin polytopes. Such polytopes were introduced by Gluskin [2] to
prove that the diameter of Minkowski compactum is of the order Ω(n).
These polytopes are constructed as the absolute convex hull of N (n)
independent vectors uniformly distributed over S n−1 and a few deterministic unit vectors. Such construction, however, cannot be adopted
to prove Theorem 1.1. Indeed, an argument based on measure concentration shows that if x1 , . . . , xN , x01 , . . . , x0N are independent random
vectors uniformly distributed over S n−1 and N ≥ n, then with high
probability
d(unc.conv(x1 , . . . , xN ), unc.conv(x01 , . . . , x0N )) ≤ C,
making it impossible to achieve distances greater than O(1). Moreover,
if N/n → ∞, then this distance tends to 1, which does not allow to
prove the doubly exponential complexity bound for distances of order
O(1) either.
To avoid the problems arising in attempts to use the standard construction of the Gluskin polytopes, we give up on the assumption that
the random vectors are uniformly distributed on the sphere. Instead,
we fix number δ > 0 and N ∈ N depending on the desired distance
and consider independent random sets I1 , . . . , IN ⊂ [n] uniformly chosen among the sets of cardinality δn. Here and below, we assume for
simplicity that the numbers δn, n/2 etc. are integer. Alternatively,
one can take the integer part of these numbers. For each l ∈ [N ], set
6
MARK RUDELSON
xl =
P
i∈Il
ei . The random convex body K will be defined as
K = K(I1 , . . . , IN )
= abs.conv unc.conv(x1 , . . . , xN ),
√
δnB1n , δ
√
nB2n
,
Here the scaled copies of B1n and B2n appear only for technical reasons, and the main role is played by the unconditional convex hull of
x1 , . . . , xN . The main advantage of this construction is that the distance between two independent copies of such bodies can be large and
can be controlled in terms of δ and N .
One of the important features of this construction is that the random points xl , l ∈ [N ] are defined via random sets Il of a fixed cardinality. This means that the coordinates of xl are P
not independent.
n
An alternative definition of random vertices yl =
i=1 νi,l ei , where
νi,l , i ∈ [n], l ∈ [N ] are independent Bernoulli random variables taking value 1 with probability δ would have been much easier to work
with. Yet, with such definition, P (ν1,1 = · · · = νn,1 = 1) = δ n , which
is only exponentially small in n. This would have made the doubly
exponential bound for probability unattainable.
We will show in Section 4 that the distance between two independent
copies of the polytope K is large with probability close to 1. This will
allow us to derive Theorem 1.1 by an application of the union bound.
The large deviation and small ball probability estimates instrumental
for the proof of the main result of Section 4 are obtained in Section 3.
3. Small ball probability and large deviation bounds
for the linear image of a random vector
We start with establishing a concentration estimate for random quadratic forms similar to the Hanson–Wright inequality. The following
Lemma is based on Theorem 1.1 [7].
Lemma 3.1. Let J be a random subset of [n] of size
P m <Tn uniformly
chosen among all such subsets. Denote by RJ = j∈j ej ej the coordinate projection on the set J. Let Y = (ε1 , . . . , εn ) be the vector whose
coordinates are independent symmetric ±1 Bernoulli random variables.
Then for any n × n matrix A and any t > 0,
P Y T RJ ARJ Y − EY T RJ ARJ Y ≥ t
^ t t2
.
≤ 2 exp −c
kAk
m kAk2
COMPLEXITY OF THE SET OF UNCONDITIONAL CONVEX BODIES
7
Proof. Let us separate the diagonal and off-diagonal terms. We have
P Y T RJ ARJ Y − EY T RJ ARJ Y ≥ t
!
!
n
X
X
X
m
t
t
≤P ajj −
ajj ≥
+P εj εk ajj ≥
2
2
n
j=1
j∈J
j∈J,j6=k
=: p1 + p2 .
We estimate p1 and p2 separately. To estimate p1 , consider a function
F on the permutation group Πn defined by
m
X
F (π) =
aπ(j),π(j) .
j=1
For k < n, denote by Al the algebra of subsets of Πn , whose elements
are the sets of permutations π for which π(1), . . . , π(l) is the same. Let
X0 (π) = EF (π), and for l ≤ m, set Xl = E[F (π) | Al ]. The sequence
X0 , . . . , Xm defined this way is a martingale with martingale differences
|Xl+1 − Xl | ≤ max |ajj | ≤ kAk .
j∈[n]
Hence, by Azuma’s inequality
ct2
.
p1 = P (|Xm − X0 | ≥ t/2) ≤ 2 exp −
m kAk2
The estimate for p2 follows directly from Theorem 1.1 [7]. Indeed, let
YJ be the coordinate restriction of the vector Y to the set J. Similarly,
let AJ be the square submatrix of the matrix A whose rows and columns
belong to J. Denote by ∆J the diagonal of AJ . Then
p
√
kAJ − ∆J k ≤ 2 kAk and kAJ − ∆J kHS ≤ |J|·kAJ − ∆J k ≤ 2 m kAk .
Therefore, by Theorem 1.1 [7] we have
p2 = P |YJT (AJ − ∆J )YJ | ≥ t/2 = EJ P |YJT (AJ − ∆J )YJ | ≥ t/2 | J
^ t t2
≤ 2 exp −c
.
kAk
m kAk2
The small ball probability bound follows immediately from Lemma
3.1.
Corollary 3.2. Let B be an n × n matrix. In the notation of Lemma
3.1,
!
r
m
m kBk4HS
P kBRJ Y k2 ≤
kBkHS ≤ 2 exp −c 2 ·
.
2n
n
kBk4
8
MARK RUDELSON
m
Proof. We apply Lemma 3.1 with A = B T B, and t = 2n
kBk2HS . In
tr(A) = 2t. Then the left hand side of the
this case, EY T RJ ARJ Y = m
n
inequality above can be bounded by
"
!#
m kBk4HS ^ m kBk2HS
exp −c
.
·
·
n2 kBk4
n kBk2
To derive the corollary, note that kBk2HS ≤ n kBk2 , so the first term
in the minimum is always smaller than the second one.
Lemma 3.1 can be also applied to derive the large deviation inequality for kBRJ Y k2 . However, the bound obtained this way will not be
strong enough for our purposes. To prove the large deviation estimate
we employ a different technique.
Lemma 3.3. Let J be a random subset of [n] of size
P m < n uniformly
chosen among all such subsets. Denote by RJ = j∈j ej eTj the coordinate projection on the set J. Let Y = (ε1 , . . . , εn ) be vector whose
coordinates are independent symmetric ±1 Bernoulli
random variables.
p
Then for any n × n matrix B and any t > 4m/n · kBkHS ,
ct2
P (kBRJ Y k2 ≥ t) ≤ 2 exp −
.
kBk2
Proof. Condition on the set J first. Note that kBRJ k ≤ kBk. Applying
Talagrand’s convex distance inequality [8], we obtain
s2
P (| kBY k2 − M | ≥ s | J) ≤ 2 exp −
,
2 kBk2
where M is the median of kBRJ Y k2 . Since M ≤ (E kBRJ Y k22 )1/2 =
kBRJ kHS , the previous inequality implies
ct2
.
(3.1)
P (kBRJ Y k2 ≥ t + kBRJ kHS | J) ≤ 2 exp −
kBk2
Set A = B T B. To finish the proof, we have to obtain a large deviation
bound for the random variable
X
U := kBRJ k2HS = tr(RJT ARJ ) =
ajj
j∈J
depending on J. The set J is chosen uniformly from the sets of cardinality m, so the elements of J are not independent.
To take advantage of independence, let us introduce auxiliary random variables. Let δ1 , . . . , δn be independent Bernoulli {0, 1}
Pnrandom
variables taking value 1 with probability 2m/n. Then P ( j=1 δj <
COMPLEXITY OF THE SET OF UNCONDITIONAL CONVEX BODIES
9
m) < 1/2. Chernoff’s inequality provides more precise bound for this
probability, but this estimate would suffice for our purposes. Set
Z = F (δ1 , . . . , δn ) =
n
X
δj ajj .
j=1
Since maxj∈[n] ajj ≤ kAk, we derive from Bernstein’s inequality that
−cτ
P (Z > τ ) ≤ exp
kAk
for any τ ≥ 4m
· tr(A) = 2EZ.
n
Compare the random variables U P
and Z. Notice that the random
variable Z conditioned on the event m
j=1 δj = m has the same distri0
00
bution as U . Also, for any m < m ,
P (Z > τ |
m
X
0
δj = m ) ≤ P (Z > τ |
j=1
m
X
δj = m00 ).
j=1
This observation allows to conclude that for any τ ≥
!
m
X
X
P Z>τ |
δj = m ·
P (δj = m0 )
≤
P
Z>τ |
m0 ≥m
· tr(A),
m0 ≥m
j=1
X
4m
n
m
X
!
0
0
· P (δj = m ) ≤ exp
δj = m
j=1
−cτ
kAk
.
Thus,
P (U > τ ) = P
Z>τ |
m
X
!
δj = m
j=1
≤
!−1
m
X
−cτ
−cτ
1 − P(
δj < m)
exp
≤ 2 exp
.
kAk
kAk
j=1
Combining
this with inequality (3.1), we obtain that for any t >
p
4m/n · kBkHS ,
P (kBRJ Y k2 ≥ 2t)
≤ EJ P (kBRJ Y k2 ≥ t + kBRJ kHS | kBRJ kHS ≤ t) + P (kBRJ k2HS > t2 )
−ct2
≤ 3 exp
.
kBk2
10
MARK RUDELSON
4. Distance between unconditional random polytopes
We follow the classical scheme of estimating the distances developed
for Gluskin’s polytopes, see e.g. [5]. Fix an n × n matrix V ∈ GL(n).
Denote the singular values of V by kV k = s1 (V ) ≥ s2 (V ) ≥ · · · ≥
sn (V ) > 0. For the Banach–Mazur distance estimate, we can normalize
V by assuming sn/2 (V ) ≥ 1 and sn/2 (V −1 ) ≥ 1. Let K and K 0 be
independent random unconditional convex bodies, and let Y be a vertex
of K. We start with estimating the probability that V Y ⊂ dK 0 for
some d > 1. For the standard Gluskin polytopes, such estimate is
obtained by volumetric considerations. In our setting, this argument
is unavailable, and we use the results of Section 3 instead.
Proposition 4.1. Let δ > Cn−1/2 , and let N = exp(cδ 2 n). For l ∈
[N ], let Il ⊂ [n] be a set of cardinality |Il | = m = δn. Define a convex
body K̃ by
√
√
√ n
IN
I1
K̃ = K̃(I1 , . . . , Il ) = abs.conv
δnB2 , . . . , δnB2 , δ nB2 .
Let J ⊂ [n] be a random subset of cardinality δn uniformly distributed
in [n], and let ε1 , . . . , εn be independent symmetric ±1 Bernoulli random variables. Set
X
Y =
εj ej .
j∈J
Then for any√linear operator V : Rn → Rn with sn/2 (V ) ≥ 1 and
log kV k ≤ 1/ δ,
c
P VY ∈ √
K̃ ≤ exp(−c0 δ 2 n).
δ log kV k
Proof. Let
V =
n
X
sj ui vi|
i=1
be the singular value decomposition of V . Then there exists an interval
I = [i1 , i2 ] ⊂ [1, n/2] of cardinality
c0 n
i2 − i1 = |I| =: r ≥
log kV k
such that si1 /si2 ≤ 2. Indeed, otherwise we would have
n/(2r)
kV k ≥
Y s(k−1)r+1
≥ 2n/(2r) > kV k
s
kr
k=1
COMPLEXITY OF THE SET OF UNCONDITIONAL CONVEX BODIES
11
provided that the constant c0 is chosen small enough. Set
X
X
Q=
si ui vi| and P =
ui u|i
i∈I
i∈I
Since the ratio of the maximal and the minimal singular values of Q
does not exceed 2, the operator Q satisfies
r kQk2 ≥ kQk2HS ≥ (r/4) kQk2 .
Note that for any α > 0, V Y ∈ αK̃ implies
QY = P QY = P V Y ∈ αP K̃.
Corollary 3.2 applied to B = Q yields
m √
P (kQY k2 ≤ si2 · c δr) ≤ exp −c 2 r2 ≤ (−c̃δ 2 n),
n
where the last inequality
√ follows from the definition of r and the assumption log kV k ≤ 1/ δ.
For l ∈ [N ], set El = span{P ej : j ∈ Il }, and let PEl be the
orthogonal projection onto El . Since kQk = si1 ≤ 2si2 ,
√
kPEl Qk ≤ 2si2 and kPEl QkHS ≤ 2si2 · kPEl kHS ≤ 2si2 · δn.
√
√
Applying Lemma 3.3 with t = Csi2 δ n > 2 δ kPEl QkHS , we get
√
2
P (kPEl QY k2 ≥ Csi2 δ · n) ≤ e−c̃δ n .
Let Ω be the event that
√
(1) kQY k2 ≥ si2 · c δr and
√
(2) kPEl QY k2 ≤ Csi2 · δ n for any l ∈ [N ].
The previous estimates show that
P (Ωc ) ≤ e−c̃δ
(4.1)
2n
+ N e−c̃δ
2n
0 2
≤ e−c δ n ,
where we used the assumption on N with a sufficiently small constant c.
Since P B2Il ⊂ B2n ∩ El , for any y ∈ Rn , we have
max hQy, xi = max hQy, P xi ≤ max
hQy, ui = kPEl Qyk2 .
n
I
x∈B2l
I
x∈B2l
u∈B2 ∩El
Assume that Ω occurs. Conditions (1) and (2) imply
p
√
(4.2)
kPEl QY k2 ≤ c n/r · δ kQY k2
12
MARK RUDELSON
for all l ∈ [N ]. If QY ∈ αP K̃, then
kQY k22 ≤ α max |hQY , xi| = α max |hQY , xi|
x∈P K̃
x∈K̃
!
≤ α max max
|hQY , xi| + max
√ n |hQY , xi|
√
l∈[N ] x∈ δnB Il
2
x∈δ nB2
√
√
≤ α δn max kPEl QY k2 + δ kQY k2
l∈[N ]
r
√
n
≤C
· αδ n kQY k2 ,
r
where the last inequality holds because of (4.2). Combining this with
(1) and recalling that si2 ≥ 1, we obtain
α ≥ c̃
r 1
c
·√ >√
n
δ
δ log kV k
c
if c is chosen small enough. This means that the event V Y ∈ √δ logkV
K̃
k
implies Ωc , and so, the proposition follows from estimate (4.1).
Proposition 4.1 pertains to one random vector Y . The body K =
K(I1 , . . . , IN ) contains many independent copies of Y , and we can use
this independence to derive the desired doubly exponential bound for
probability. If K and K 0 are independent convex bodies, denote by P K
the probability with respect to K conditioned on K 0 being fixed.
Corollary 4.2. Let δ > Cn−1/2 , and let N = exp(cδ 2 n). Let Il , l ∈ [N ]
be independent random subsets of [n] uniformly
chosen among the sets
P
of cardinality δn. For l ∈ [N ] denote xl = j∈Il ej . Consider a random
convex body
√
√
K = K(I1 , . . . , IN ) = conv unc.conv(x1 , . . . , xN ), δnB1n , δ nB2n ,
and let K 0 be an independent copy of K. Assume that K 0 = K(I10 , . . . , IN0 )
S
0
satisfies N
l=1 Il = [n].
n
Let V : R → Rn be a linear operator such that sn/2 (V ) ≥ 1. Then
c0
0
K ≤ exp − exp(cδ 2 n) .
PK V K ⊂ √
δ log(1/δ)
Proof. Denote for shortness
α= √
c
.
δ log(1/δ)
COMPLEXITY OF THE SET OF UNCONDITIONAL CONVEX BODIES
13
√
√
Assume that V √
K ⊂ αK 0 . Since K ⊃ δ nB2n and K 0 ⊂ δnB2n , we
have kV k ≤ α/ δ, which implies
log kV k ≤ C log(1/δ).
The condition on K 0 implies that K 0 ⊂ K̃(I1 , . . . , IN ), where the last
set is defined in Proposition 4.1.
Let εj,l , j ∈ [n], l ∈ [N ] beP
independent symmetric ±1 random
variables. Then the vectors Yl = j∈Il εj ej are contained in K. Hence,
by Proposition 4.1,
N
N
Y
0
P (V K ⊂ αK ) ≤
P (V Yl ∈ αK 0 ) ≤ exp(−cδ 2 n)
l=1
≤ exp − exp(cδ 2 n)
as required.
Our main technical result, Theorem 4.3 below, will imply Theorem 1.1 almost immediately.
q
Theorem 4.3. Let δ > C logn n , and let N = exp(cδ 2 n). For l ∈ [N ],
let IlP⊂ [n] be a set of cardinality |Il | = δn. For l ∈ [N ] denote
xl = j∈Il ej . Consider a random convex body
K = K(I1 , . . . , IN )
√
√
= abs.conv unc.conv(x1 , . . . , xN ), δnB1n , δ nB2n ,
and let K 0 be an independent copy of K. Then
c1
0
2
P K,K 0 d(K, K ) ≤
≤ exp − exp(c2 δ n) .
δ log2 (1/δ)
Proof. The proof follows the general scheme developed by Gluskin. Fix
K 0 = K(I10 , . . . , In0 ) such that ∪nl=1 Il0 = [n]. Let c be a constant to be
chosen later. Denote by W (K) the event
c
W (K, K 0 ) = {∃V : Rn → Rn sn/2 (V ) ≥ 1 and V K ⊂ √
K 0 }.
δ log(1/δ)
We start with proving the following
Claim. P K (W (K, K 0 )) ≤ exp − exp(cδ 2 n) .
c0
Set ε = 2 log(1/δ)
, where c0 is the constant from Corollary 4.2. By
Corollary 8 [5], the set of all operators V 0 : Rn → Rn such that
14
MARK RUDELSON
sn/2 (V ) ≥
cardinality
√
√
√
δ/ε and V 0 ( δnB1n ) ⊂ K 0 possesses an δ-net N 0 of
|N 0 | ≤
c
√
δ
n2
·
!n
√
vol(K 0 / δn)
2
≤ Cn ,
n
vol(B2 )
where we used K 0 ⊂ δnB1n to obtain
√ the0 last inequality. n2 Hence,
0
W (K, K ) possesses an ε-net N = (ε/ δ)N of cardinality C .
Assume now that there exists √
an operator V : Rn → Rn such
that sn/2 (V ) ≥ 1 and V K ⊂ (ε/ δ)K 0 . Let V0 ∈ N be such that
kV − V0 k < ε. Then
kV0 : K → K 0 k ≤ kV : K → K 0 k + kV − V0 : K → K 0 k
√
√
ε
≤ √ + V − V0 : δnB2n → δ nB2n δ
2ε
c0
≤√ =√
.
δ
δ log(1/δ)
This means that
c0
K 0)
δ log(1/δ)
c0
K 0)
≤ |N | · max P K (V0 K ⊂ √
V0 ∈N
δ log(1/δ)
P K (W (K, K 0 )) ≤ P K (∃V0 ∈ N V0 K ⊂ √
Combining the bound for |N | appearing above with Corollary
4.2, we
2
show that this probability does not exceed exp −exp(cδ n) provided
q
that δ > C logn n for a sufficiently large C. This completes the proof
of the Claim.
To derive the Theorem from the Claim, note that the inequality
d(K, K 0 ) ≤ δ log2c(1/δ) guarantees the existence of a linear operator V :
Rn → Rn such that
c
kV : K → K 0 k · V −1 : K 0 → K ≤
.
2
δ log (1/δ)
Without loss of generality, we may assume that sn/2 (V ) ≥ 1 and
sn/2 (V −1 ) ≥ 1. This means that the event W (K, K 0 ) occurs with operator V , or W (K 0 , K) occurs with V −1 .
Also,
!
N
[
P K0
Il0 6= [n] ≤ n(1 − δ)N ≤ exp − exp(cδ 2 n) .
l=1
COMPLEXITY OF THE SET OF UNCONDITIONAL CONVEX BODIES
15
Therefore,
P K,K 0 d(K, K 0 ) ≤
≤ EK 0 P K
c
δ log2 (1/δ)
!
N
[
W (K, K 0 ) |
Il0 = [n] + P K 0
l=1
+ EK P K 0
W (K 0 , K) |
N
[
N
[
!
Il0 6= [n]
l=1
!
Il = [n]
l=1
+ PK
N
[
!
Il 6= [n]
l=1
≤ 4 exp − exp(cδ 2 n) .
Theorem 4.3 is proved.
Proof of Theorem 1.1. Let
δ=
c1
,
t log (1 + t)
2
where c1 is the constant from Theorem 4.3. Choosing the constant c̃ in
the formulation q
of Theorem 1.1 sufficiently small, we ensure that the
condition δ > C logn n holds, and Theorem 4.3 applies. Set
c
c
2 2
n
,
M = exp exp
δ n = exp exp 2 4
4
t log (1 + t)
with some constant c > 0.
Consider M independent unconditional random convex bodies K1 , . . . , KM
which are constructed as in Theorem 4.3. By this theorem and the
union bound,
c
P ∃m, m̄ ∈ [M ] m 6= m̄, d(Km , Km̄ ) ≤
δ log2 (1/δ)
c
2 2
≤ M 2 · exp − exp(c2 δ 2 n) ≤ exp − exp
δ n .
2
unc
This inequality implies that Kn contains a t-separated set of cardinality at least M .
We now pass to the proof of Corollary 1.2. Since this corollary follows
from Theorem 1.1 and [4], we will provide only a sketch of the proof
instead of a complete argument.
Proof of Corollary 1.2 (sketch). Fix m, n ≤ m ≤ N . Following the
proof of Theorem 1.1 [4], we estimate of the cardinality of a special 2net in the set of all n-dimensional projections of m-dimensional sections
16
MARK RUDELSON
of the simplex ∆N ⊂ RN . By Lemmas 3.1, 3.2 [4], the sections of the N dimensional simplex can be encoded by the points of the Grassmanian
GN√
+1,m so that an ε-net A1 on the Grassmanian corresponds to a (1 +
εm N + 1)2 -net N1 in the set of the sections of the simplex in the
Banach-Mazur distance. Similarly, by Lemma 3.3 [4], for any K ∈ N1
we can encode all n-dimensional projections of K by points of the
Grassmanin
that an ε-net A2 on this Grassmanian corresponds
√ Gm,n so
2
to (1+εm N + 1) -net in the set of the projections of K in the BanachMazur distance. Combining these two results,
we see that the points
√
of A1 × A2 correspond to some (1 + εm N + 1)4 -net in the set of the
projections of of the sections of the simplex. The nets A1 , A2 can be
chosen so that
C̃[(N +1)m+mn]
C
C
0 2
≤ exp C N log
|A1 × A2 | ≤
.
ε
ε
√
Choosing ε such that (1 + εm N + 1)4 = 2, we derive that there exists
a 2-net Mm in the set of all n-dimensional projections of m-dimensional
sections of ∆N of cardinality
|Mm | ≤ exp(cN 2 log N ).
Setting M = ∪N
m=n Mm , we obtain a 2-net in the set of all n-dimensional
projections of sections of ∆N satisfying a similar estimate.
If any n-dimensional unconditional convex body can be approximated by a projection of a section of ∆N within the distance d, then
M is a (2d)-net in the set Knunc . This means that the cardinality of
any (2d)2 -separated set in Knunc does not exceed exp(cN 2 log N ).
Let us show that this implies the desired lower bound on d. Assume
that
1/4
n
n
−1
0
d≤c
· log
,
log N
log N
where the constant c0 > 0 will be chosen later. If c0 is sufficiently small,
then the inequality
(2d)2 ≤ c̃n1/2 log−5/2 n
holds, and so Theorem 1.1 applies. By this theorem, there exists a
(2d)2 -separated set of cardinality at least
cn
.
exp exp
(2d)4 log4 ((2d)2 + 1)
COMPLEXITY OF THE SET OF UNCONDITIONAL CONVEX BODIES
17
Combining this with the upper estimate for this cardinality proved
above, we obtain
cn
≤ 3 log N.
4
4
(2d) log ((2d)2 + 1)
This contradicts our assumption on d if the constant c0 is chosen sufficiently small.
5. Complexity of the set of completely symmetric bodies
In this section, we prove Proposition 1.3 establishing the upper
bound on the cardinality of a t-separated set in the set of all completely symmetric bodies. Denote the set of all completely symmetric
bodies in Rn by Kcs .
Proof. Fix 1 < τ < n. Let L ∈ N be the smallest number such that
(5.1)
nτ −L < 1 − τ −1 .
Denote by Y the set of all non-decreasing functions ψ : [L] → [n] such
that ψ(1) < ψ(n). Note that
n+L
|Y | ≤
≤ (n + L)L .
L
Define a function Φ : Kcs → RY by
L
X
X −l
τ
Φψ (K) = ej ,
l=1
ψ(l−1)<j≤ψ(l) ψ ∈ Y, K ∈ Kcs
K
where we use the convention ψ(0) = 0.
Assume that K, D ∈ Kcs . We will prove that if
τ −1 Φψ (D) ≤ Φψ (K) ≤ τ Φψ (D)
for all ψ ∈ Y , then
(5.2)
d(K, D) ≤ τ 6 .
To this end, take any vector x ∈ Rn such
Pnthat kxkK = 1 and x1 ≥
· · · ≥ xn ≥ 0. Define the vector y =
j=1 yj with coordinates yj
−l
taking values in the set {0} ∪ {τ , l ∈ [L]} so that
• yj ≤ xj < τ yj , if |xj | ≥ τ −L ;
• yj = 0, if xj < τ −L .
Then
X
−L
−1
kxkK ≤ τ kykK + xj e j ≤ τ kykK + nτ ≤ τ kykK + 1 − τ ,
xj <τ −L
K
18
MARK RUDELSON
where we used (5.1) in the last inequality. Hence, 1 = kxkK ≤ τ 2 kykK .
Also, the inequalities Φψ (K) ≤ τ Φψ (D), ψ ∈ Y imply kykK ≤ τ kykD .
Combining this with kykD ≤ kxkD , we obtain kxkK ≤ τ 3 kxkD , and
reversing the roles of K and D, we derive
τ −3 kxkD ≤ kxkK ≤ τ 3 kxkD ,
which implies (5.2).
Define now a new function Θ : Kcs → RY by setting Θψ (K) =
log Φψ (K). Then Θ(Kcs ) ⊂ [− log(τ 2 ), log n]Y . Hence, there exists a
(log τ )-net N ⊂ Θ(Kcs ) in the k·k∞ -norm of cardinality
|Y |
log n
|N | ≤
+2
.
log τ
For any x ∈ N , choose a body Kx ∈ Kcs such that Θ(Kx ) = x and
consider the set M = {Kx : x ∈ N }. Then for any K ∈ Kcs , there
exists Kx ∈ M with kΘ(K) − Θ(Kx )k∞ ≤ log τ . This means that
for any ψ ∈ Y , τ −1 Φψ (Kx ) ≤ Φψ (K) ≤ τ Φψ (Kx ) for all ψ ∈ Y , and
by (5.2), d(K, Kx ) ≤ τ 6 . Thus, we constructed a τ 6 -net M in Kcs of
cardinality
|Y | (n+L)L
log n
log n
+2
≤
+2
.
|M| ≤
log τ
log τ
n
Assume now that τ ≥ 21/12 . Then, by (5.1), L < c log
, and the
log τ
previous inequality implies
log2 n
|M| ≤ exp exp C
.
log τ
By the multiplicative triangle inequality, the same inequality holds for
the cardinality of any τ 12 -separated set in Kcs . To derive the statement
of the Proposition, set τ = t1/12 .
Remark 5.1. The same proof works for all values t > 1. However,
in the case 1 < t ≤ 2 the estimate of L in (5.1) in terms of n and t
is different, which leads to a different estimate of the cardinality of a
t-separated set.
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